Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6792416 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #38904 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,16)(4,21)(5,26)(6,31)(7,13)(8,15)(9,19)(10,24)(11,28)(12,20)(14,32)(17,36)(18,35)(22,43)(23,50)(25,38)(27,40)(29,41)(30,42)(33,39)(34,49)(37,53)(44,47)(45,59)(46,57)(48,56)(51,61)(52,55)(54,60)(58,62)(63,64);(1,3,7,19)(2,9,13,16)(4,22,25,49)(5,18,28,47)(6,32,20,8)(10,36,39,57)(11,35,26,44)(12,14,31,15)(17,33,46,24)(21,43,38,34)(23,51,54,62)(27,29,48,30)(37,59,55,64)(40,42,56,41)(45,52,63,53)(50,61,60,58);(1,4)(2,10)(3,17)(5,23)(6,22)(7,25)(8,33)(9,34)(11,37)(12,36)(13,39)(14,38)(15,21)(16,43)(18,45)(19,46)(20,49)(24,32)(26,55)(27,51)(28,54)(29,53)(30,52)(31,57)(35,58)(40,59)(41,50)(42,60)(44,61)(47,63)(48,62)(56,64);(1,5)(2,11)(3,18)(4,23)(6,27)(7,28)(8,30)(9,35)(10,37)(12,40)(13,26)(14,42)(15,41)(16,44)(17,45)(19,47)(20,48)(21,50)(22,51)(24,53)(25,54)(29,32)(31,56)(33,52)(34,58)(36,59)(38,60)(39,55)(43,61)(46,63)(49,62)(57,64);(1,6,32,19,7,20,8,3)(2,12,14,16,13,31,15,9)(4,22,24,46,25,49,33,17)(5,27,29,47,28,48,30,18)(10,36,38,43,39,57,21,34)(11,40,42,44,26,56,41,35)(23,51,53,63,54,62,52,45)(37,59,60,61,55,64,50,58)Polycyclic presentation:
2^9|b^2=i,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699613);